In recent years, quantum technology has flourished and quantum imaging technology has developed. Ghost imaging is one of the hot new imaging technologies. Ghost imaging utilizes properties of quantum entanglement to achieve nonlocal image transmission. Due to the existence of correlation characteristics, its anti-interference ability, weak optical imaging capability, and encryption capability have strong advantages, compared to traditional optics. However, since ghost imaging is still transmitted through the optical path, the image quality is inevitably affected by the quality of the optical path, and light is transmitted through the air, so atmospheric turbulence is inevitably an important factor affecting the associated imaging.

As a new imaging method, associative imaging has attracted wide attention in the field of image transmission [1-9]. Ghost-imaging techniques continue to evolve, from entangled photon pair [1] to thermal sources [2], from traditional ghost imaging to computational ghost imaging (CGI) [3], and many improved methods such as compressed sensing (CS) [4] and differential correlation imaging [5] have also been proposed. With the improvement of image quality, the influence of turbulence on imaging has attracted more and more attention. In reference [6], a random phase screen was applied to simulate the phase disturbance of turbulence. In reference [7], a thermo-optical ghost-imaging experiment was carried out to verify that it is turbulence-free. In reference [10], the visibility and quality of the ghost image through monostatic turbulence and bistatic turbulence were discussed. In reference [11], numerical calculations to demonstrate atmospheric effects on ghost imaging of a double slit were studied. In reference [12], the spatial resolution, image contrast, and SNR were calculated for ghost imaging through turbulence on rough-surfaced targets. However, the above references rarely research the intensity fluctuation brought by turbulence in a system for compressed-sensing computational ghost imaging (CSCGI).

A variety of intensity-fluctuation turbulence channel models, e.g. log-normal, negative exponential, and gamma-gamma, were proposed to analyze the effects of turbulence on a free-space optical (FSO) communication system [13-16]. Among the above models, the gamma-gamma intensity-fluctuation model has been most widely applied in analyzing the BER of a communication (transmission) system, since it can fit a broad range of turbulence conditions (from weak to strong) [17]. In CGI, the key to receiving the transmitted images is to obtain a specific intensity value from a bucket detector. The value is the sum of intensity in the receiving area of the bucket detector. The PSNR and BER performance are directly related to the sum of intensity, as discussed above. Obviously, obtaining the received intensity in CGI is similar to that of an FSO system. Based on this common ground, it may be reasonable to apply the gamma-gamma intensity fluctuation model to analyze the PSNR and BER performance of the CGI system.

The rest of this paper is organized into three sections: The theories of CSCGI and atmospheric turbulence channel are expressed in Sections 2.1 and 2.2 respectively. Then, the scheme for the CSCGI system under GG atmospheric turbulence channel is discussed in Section 2.3. Accordingly, in Section III, PSNR and BER are simulated to measure the performance of the proposed scheme. The PSNR and BER performance are determined by the refractive-index structure parameter , the transmission distance d, and the sampling rate N, as further discussed in Sections 3.1 to 3.3. At last, the conclusion of the whole work is presented in Section VI.

Ghost imaging, also known as correlated imaging or two-photon imaging, is a quantum or classical correlation feature based on light-field fluctuations. By measuring the intensity correlation function between the reference light field and the target-detection light field, a novel imaging technique that acquires target image information nonlocally is achieved. Correlated imaging is different from general, classical optical imaging, and can be independent of the speed of light; this is an important feature that distinguishes quantum optics from classical theory. It is a new type of imaging technology, but image resolution and contrast are still the most important technical objectives.

Traditional correlation imaging requires two optical paths, causing problems such as experimental difficulties and occupying a large space. To solve these problems, Shapiro proposed computational ghost imaging (CGI) [18], and Bromberg et al. improved CGI. Instead of rotating the ground glass, we applied a spatial light modulator to simplify the two light paths into one [19]. The principle of computational ghost imaging is shown in Fig. 1. The most important device in computational ghost imaging is the digital micromirror device (DMD), which is placed on the optical path where the object is located and controlled to produce a series of photos with random intensity or random phase. A bucket detector with no spatial resolution is used to receive the total intensity of the light field after the light has passed through the object [20].

While the computational ghost imaging is running, a series of random modulation matrices ϕ_i (x, y) are loaded onto the DMD. When the laser is irradiated on the DMD, the modulated light is reflected onto the object D (x, y). Then the object is imaged, and the light transmitted through it is detected by the bucket detector (only when the light has passed through the object will the total intensity, without any resolution, be detected), and the total intensity I_i is calculated as shown in Eq. (1). Here the index i indicates sample number. After sampling z times, the modulation matrix and the value from the bucket detector are correlated, and the image of the object can be recovered. The correlation function C_CGI (x, y) is expressed in Eq. (2).

Here 〈I〉 is the average value of light intensity, and N is the number of samples.

To improve the quality of the reconstructed image, various algorithms are introduced on the basis of ghost imaging, and compressed sensing is one of them. Compressed sensing, also known as compressed sampling or sparse sampling, is a technique for finding sparse solutions for underdetermined linear systems. Compressed-sensing theory utilizes the sparse characteristics of the signal, and the amount of effective information required is much less than the sampling amount required by the Nyquist sampling theorem, which reduces the amount of information collected, saves signal acquisition time, and still reconstructs high-precision images. Low sampling and high reconstruction have been achieved. Combining compressed sensing and ghost imaging, the number of samples has been effectively reduced and the quality of image reconstruction have been effectively improved [21].

The formula for compressed sensing is:

x is the original one-dimensional signal of length N. φ is the observation matrix, used to project the original high-dimensional signal x into the low-dimensional space; y is the one-dimensional measurement of length M. The measured value y and the measurement matrix φ are known, and the original signal x is obtained by solving the equation. In our scheme, x is a sparse signal.

In computational ghost imaging, the intensity I_i is recorded by the bucket detector as expressed in Eq. (4). Here “/” refers to the symbol of divisibility, and I_i is equal to the measured value y in Eq. (3), while the random matrix ϕ_i(x, y) used in each measurement is stretched into a row vector as the observation matrix φ of Eq. (3), and the intensity of the object D(x, y) for imaging is equal to the intensity of the original signal x in Eq. (3).

The upper atmosphere’s pressure and temperature are affected by wind and other factors. Accordingly, the refractive index of an optical path varies, and atmospheric turbulence occurs. This turbulence results in intensity fluctuations of the received signal, and increases the BER.

There are two methods to simulate turbulence: The first method is to study the effect of atmospheric turbulence on light intensity, i.e. the distribution model of light intensity. The second method is to generate Kolmogorov phase screens, which are based on the Kolmogorov spectrum.

In ghost imaging, there is a bucket detector receiving the total intensity of the transmission image, while the resolution and phase of each pixel in the image is not mainly a concern. As a result, we apply the first method, namely the distribution model, to simulate turbulence. The log-normal and gamma-gamma models are classical models to describe light-intensity distribution. The former is valid for weak turbulence, while the gamma-gamma distribution could be a better choice for a broad range of turbulences(from weak to strong), because both large-scale and small-scale intensity fluctuations can be approximated by gamma distributions.

The gamma-gamma distribution is as follows [17]:

In Eq. (6), and is the refractive-index structure parameter. The degree of atmospheric refractive-index fluctuation is measured by , which varies with altitude and wind speed. In the ITU (International Telecommunication Union) -R H-V model, the relationship between and altitude h is expressed in Eq. (7) and Fig. 2 [22, 23]. The conditions are set as v_RMS = {11, 21, 31} and :

It can be seen from Fig. 2 that is approximately invariant with h when h is larger than 1000 m, and also considered approximately invariant with v_RMS when h is lower than 4000 m.

In this paper, transmission in the horizontal path is researched, so is practically constant. The values of for the different turbulence regimes are as follows:

A schematic diagram of image transmission in compressed-sensing computational ghost imaging with the added turbulence model is shown in Fig. 3. After being modulated by the modulation matrix ϕ_i of the DMD, the laser passes through the object O(x, y) , then passes through the turbulent air mass φ_i(x, y), and is finally received by the bucket detector. After the correlation calculation, the original image is reconstructed. The compressed-sensing computational ghost-imaging transmission based on the gamma-gamma turbulence channel model (GG-CSCGI) is expressed in Fig. 3.

The received light intensity is obtained from Eq. (8), considering the gamma-gamma turbulence in Eq. (5):

Here ϕ_i(x, y) is the i^th modulation matrix, O(x, y) is a function used to describe the light intensity of the object, and φ_i(x, y) is multiplicative noise with a gamma-gamma distribution.

Next we bring the turbulence receiving function of the bucket detector into the compressed-sensing computational ghost imaging of Eqs. (4) and (8) to get the intensity array of the reconstructed image, as shown in Eq. (9):

Here z is the total number of samples, ϕ_i(x, y) is a row vector that is stretched by the modulation matrix for each sample, and I_i the values received by the bucket detector, which can be calculated using Eq. (8). Researching how to set appropriate parameters in the GG-CSCGI system is the key to eliminating the negative influence of atmospheric turbulence and improving imaging quality. The PSNR and BER performance are discussed in the following section.

In the simulation, a 64 × 64-pixel binary image (letters “USST”, white on black) for correlation imaging has been selected, and the multiplicative gamma-gamma model of turbulence has been introduced. Three parameters―the transmission distance d, refractive-index structure parameter , and sampling rate N (which is defined as the ratio of sampling times to the total number of pixels)―are varied, to study the effect of atmospheric turbulence on correlated CSCGI.

In the simulation, we use the gamrnd function in Matlab to generate the gamma-gamma RV matrix, according to Eq. (5). The parameters α and β are calculated by Eq. (6), and the RV matrix is used as the multiplicative factor added into the correlation imaging system.

To detect the quality of the reconstructed image, two objective evaluation indicators, PSNR and BER, are used. Since the original image is a 64 × 64 binary image with only 4096 points, the image should be repeated to simulate enough points and calculate a sufficiently low BER. PSNR is an objective evaluation indicator for image quality, most commonly used in image processing. After compression, transmission, and other operations, the processed image usually differs from the original image. To measure the quality of the processed image, the PSNR value is usually used to determine whether the compressed transmission process conforms to the standard. PSNR is typically used for comparison of the maximum of the signal to the background noise. The larger the PSNR value between the two images, the less the distortion, and the more similar the two images are. PSNR can be calculated by Eqs. (10) and (11) as follows:

Here MSE is the mean square error between the original image and the output image, and MAX_I indicates the maximum value of the image color. In this experiment binary images are transmitted, so the value of MAX_I is taken to be 1.

BER is also an indicator of the accuracy and reliability of a transmission system. It is defined as the ratio of the number of error bits to the number of all transmitted bits, as expressed in Eq. (12):

In this article, Eq. (12) is specified as:

N_Yxy-X_xy is the number of input and output pixels with nonzero difference, and x × y is the total number of pixels.

In Fig. 4, the relationship between the PSNR and the BER of a GG-CSCGI system is expressed. To eliminate the inherent error of the associated imaging, N is set as the full sampling rate, namely 100%. Let the peak power of the transmitted binary signal be 1, and the noise power be the variance of the multiplicative turbulence noise; thus the PSNR can be obtained. In addition, the value of PSNR can be varied by changing the key parameters of the turbulence channel, namely the refractive-index structure parameter and transmission distance d. This operation will be discussed in the following paragraphs. It can be seen from Fig. 4 that as PSNR increases, BER decreases. For PSNR values between 10 dB and 17 dB, the BER remains high (>10^-1). When the PSNR is greater than 17 dB, the BER obviously decreases. For each 1 dB increase in PSNR, the BER decreases by one order of magnitude, and it has dropped below 10^-3 when the PSNR is 30 dB.

To ensure that the associated imaging channel is only affected by turbulence, N is taken as 100%. When analyzing the influence under different transmission distances, the refractive-index structure parameter is taken as 10^-13, 10^-15, and 10^-17, and the distance between the transmitter and the receiver d is set to a series of values. Thus the according imaging effects at different d are shown in Table 1.

When and the value of d ranges from 100 to 400 m, the imaging performance significantly changes from fine to bad. Similar significant changes in the imaging performance occur for the situations of with d ranging from 1000 to 4000 m, and with d ranging from 10,000 to 70,000 m. PSNR and BER performance with d are expressed in Fig. 5.

Obviously, as d increases PSNR rises, while BER declines. Both tend to stabilize when d increases to a specific value, which varies with .

Similarly, when analyzing the influence of , N is also set to 100%. When analyzing the influence under different values of the refractive-index structure parameter, the transmission distance d is set to 100 m, 1000 m, and 10,000 m respectively, and is set to a series of values. Thus the according imaging effects for different d are shown in Table 2.

When d = 100 m and the range of is from 1 × 10^-13 to 7 × 10^-13, the imaging performance significantly changes from fine to bad. Similar significant changes from fine to bad. Similar significant changes in imaging performance occur in the situations with d = 1000 m and ranging from 1 × 10^-15 to 7 × 10^-15, and d = 10,000 m and ranging from 1 × 10^-17 to 7 × 10^-17. PSNR and BER performance with varying are expressed in Fig. 6.

The trend in Fig. 6 is opposite than that in Fig. 5: As increases, PSNR rises, while BER declines. Both tend to stabilize when increases to a specific value that varies with d.

Next the refractive-index structure parameter is taken as 1 × 10^-15 and the distance d as 2000 m, and the sampling rate N of the associated imaging is adjusted, to observe the influence of turbulence for different N.

The corresponding image effects are shown in Table 3, and the PSNR and BER performance with varying N are expressed in Fig. 7.

When N is very low, whether turbulence is considered or not, the imaging is not satisfactory. When N = 15%, the reconstructed image is noisy, almost regardless of whether turbulence is added or not. When N = 35%, both cases return an identifiable original image, but there is much noise; the reconstructed image without turbulence is better than the reconstructed image with turbulence. When N = 100%, the compressed sensing has completely restored the original image without noise, a nd t he images for turb ulent compressed sensing also present the clear letters of the original image, though a small amount of noise is still visible. In the simulation, the compressed sensing significant suppresses the negative effect of turbulence, and when N is sufficient, the image can be restored very well.

As N increases, PSNR rises, while BER declines. In addition, the PSNR of the reconstructed image without turbulence is higher than that of the reconstructed image considering turbulence, while the corresponding BER without turbulence is much lower than that considering turbulence.

In this paper, the BER performance of a compressed-sensing computational ghost-imaging system based on the gamma-gamma atmospheric turbulence channel model has been researched. The influence of light intensity for the gamma-gamma atmospheric turbulence channel on image transmission has been analyzed. Specifically, the intensity influence is mediated by the key parameters of the turbulence channel, namely the refractive-index structure parameter and transmission distance d. Simulation results have shown that the key parameters are both negatively correlated with PSNR and positively correlated with BER. The PSNR and BER approach certain constants and then are stable when either of the key parameters increases sufficiently. The whole work provides a theoretical model basis and reference for a practical ghost-imaging system transmitting through an atmospheric turbulence channel.